C₂³×#3492 #72301670

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂³×#3492

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

513125 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 8 generators on 22 points:

(3,4);(5,6);(1,2)(3,4)(5,6);(7,8,11,13)(9,10,12,14)(15,16,20,22)(17,18,19,21);(7,8)(9,10)(11,13)(12,14)(15,17,20,19)(16,21,22,18);(7,9,11,12)(8,10,13,14)(15,18,20,21)(16,17,22,19);(7,10,11,14)(8,12,13,9);(7,8)(9,10)(11,13)(12,14)(15,19,20,17)(16,21,22,18)

Conjugacy classes & character table

352 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂³×#62818, C₂³×#62773
maximal normal subgroup of	C₂³×#62818, C₂³×#62773
maximal quotient of	C₂³×#59220, C₂³×#36581

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((3,4), (5,6), (1,2)(3,4)(5,6), (7,8,11,13)(9,10,12,14)(15,16,20,22)(17,18,19,21), (7,8)(9,10)(11,13)(12,14)(15,17,20,19)(16,21,22,18), (7,9,11,12)(8,10,13,14)(15,18,20,21)(16,17,22,19), (7,10,11,14)(8,12,13,9), (7,8)(9,10)(11,13)(12,14)(15,19,20,17)(16,21,22,18));

C23×#3492 #72301670